Equivalence of mixed and nonconforming methods on general polytopal partitions. Part I: Multiscale and projection methods

TL;DR

This paper investigates the equivalence between mixed and nonconforming finite element methods on general polytopal partitions for variable diffusion problems, focusing on multiscale and projection methods, and establishing new theoretical results.

Contribution

It establishes the first-level equivalence of multiscale methods without oversampling and provides practical criteria for the equivalence of projection methods, extending previous results.

Findings

Equivalence of four multiscale approaches without oversampling

A practical criterion for primal/mixed well-posedness and equivalence

New insights into self-stabilized hybrid methods

Abstract

We study equivalence, in the context of a variable diffusion problem, between (conforming) mixed methods and (primal) nonconforming methods defined on potentially general polytopal partitions. In this first paper of a series of two, we focus on multiscale and projection methods. For multiscale methods, we establish the first-level equivalence between four different (oversampling-free) approaches, thereby broadening the results of [Chaumont-Frelet, Ern, Lemaire, Valentin; M2AN, 2022]. For projection methods, in turn, we provide a simple criterion (to be checked in practice) for primal/mixed well-posedness and equivalence to hold true. In the process, we also shed a new light on some self-stabilized hybrid methods. Part II of this work will address (general) polytopal element methods.